No. To be "in hospital" or to go "to hospital" means that you are a patient. To be "in the hospital" describes only location.
> Your description is incomplete
Your analysis is fine, but does not contradict my comment and moreover is irrelevant to the present discussion, which pertains to the subtle use of articles in English.
>>> in UK English, not so much American English
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> Your analysis is fine, but does not contradict my comment and moreover is irrelevant to the present discussion, which pertains to the subtle use of articles in English.
As you can see from the dialectal difference you've already described, what you're talking about isn't really related to "the subtle use of articles in English". American English and British English agree on how their articles are used. Where they agree on "in school" and disagree on "in hospital", you can recognize that usage in those phrases, rather than revealing some kind of subtle point about how the is used in nearly-identical contexts, is determined arbitrarily.
I wasn't aware that I was making a "subtle" point. I was joint showing that the use of articles in English is complex and hard-to-predict for non-native speakers, in contradiction to the claim (paraphrased) that "Just use the for things that are definite, duh!" Your observation that article use varies between dialects of the same language strengthens my point, insofar as encountering inconsistent use makes learning the language even harder.
> is determined arbitrarily.
I don't know what you're trying to say here. Languages vary systematically over time. Just because two different dialects, or two different languages, do things in different ways doesn't mean that those changes were arrived at arbitrarily.